Luiz Antonio Peresi

Special Identities for Quasi-Jordan Algebras

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤7 is a consequence of the three identities in degree ≤4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.

On the solvability of the commutative power-associative nilalgebras of dimension 6

Abstract We prove that commutative power-associative nilalgebras of dimension 6 over a field of characteristic ≠2,3,5 are solvable.

On the Solvability of the Five Dimensional Commutative Power-Associative Nilalgebras

We prove that commutative power-associative nilalgebras of dimension 5 are solvable.

Solvability of the ideal of all weight zero elements in bernstein algebras

We use a computer to verify that the ideal N of all weight zero elements of any (not necessarily finite dimensional) Bernstein algebra is solvable of index ≤4. We also use a computer to verify that N 2 is nilpotent of index ≤9. We give three examples of Bernstein algebras which show that various hypotheses like finite dimensionality, finitely generatedA 2 = A, are separately not enough to force N to be nilpotent.

Classification of Trilinear Operations

We use the representation theory of the symmetric group to classify up to equivalence all multilinear operations over the field ℚ of rational numbers. In the case n = 3, we obtain explicit representatives of the equivalence classes of trilinear operations From these results we obtain one-parameter families of deformations of the classical Lie, Jordan, and anti-Jordan triple products and the corresponding varieties of triple systems. For one representative of each equivalence class, we use computational algebra to study the nonassociative polynomial identities satisfied by the operation in every totally associative ternary algebra. We obtain 19 new trilinear operations for which the corresponding varieties of triple systems are defined by identities of degrees 3 and 5. For 10 of these operations we classify their obvious identities in degree 3 and their minimal identities in degree 5. Our main goal is to give new examples of classes of triple systems.

Semi-prime Bernstein algebras

In this paper we study the semiprime case. That is, we consider Bernstein algebras that do not have nonzero nilpotent ideals of index two. We prove that any such algebra is Jordan. Furthermore, under the condition that the algebra is finitely generated, we show that it must be a field. The proofs require characteristic different from two. Our work implies that nearly all (finitely generated) Bernstein algebras possess nonzero ideals which are nilpotent of index two. The only ones which do not are the fields.

Dimension Formulas for the Free Nonassociative Algebra

ABSTRACT The free nonassociative algebra has two subspaces which are closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we prove a closed formula for the dimension of the primitive elements. These results generalize the Witt dimension formula for the Lie elements in the free associative algebra.

STRUCTURE THEORY FOR THE GROUP ALGEBRA OF THE SYMMETRIC GROUP, WITH APPLICATIONS TO POLYNOMIAL IDENTITIES FOR THE OCTONIONS

In part 1, we review the structure theory of

A Nonzero Element of Degree 7 in the Center of the Free Alternative Algebra

\mathbb{F} S_n

A note on duplication of algebras

, the group algebra of the symmetric group